Abstract
We propose an online adaptive local-global POD-DEIM model reduction method for flows in heterogeneous porous media. The main idea of the proposed method is to use local online indicators to decide on the global update, which is performed via reduced cost local multiscale basis functions. This unique local-global online combination allows (1) developing local indicators that are used for both local and global updates (2) computing global online modes via local multiscale basis functions. The multiscale basis functions consist of offline and some online local basis functions. The approach used for constructing a global reduced system is based on Proper Orthogonal Decomposition (POD) Galerkin projection. The nonlinearities are approximated by the Discrete Empirical Interpolation Method (DEIM). The online adaption is performed by incorporating new data, which become available at the online stage. Once the criterion for updates is satisfied, we adapt the reduced system online by changing the POD subspace and the DEIM approximation of the nonlinear functions. The main contribution of the paper is that the criterion for adaption and the construction of the global online modes are based on local error indicators and local multiscale basis function which can be cheaply computed. Since the adaption is performed infrequently, the new methodology does not add significant computational overhead associated with when and how to adapt the reduced basis. Our approach is particularly useful for situations where it is desired to solve the reduced system for inputs or controls that result in a solution outside the span of the snapshots generated in the offline stage. Our method also offers an alternative of constructing a robust reduced system even if a potential initial poor choice of snapshots is used. Applications to single-phase and two-phase flow problems demonstrate the efficiency of our method.
Highlights
Reduced order models (ROM) aims at reducing the computational complexity and, in turn, the simulation time of large-scale dynamical systems by approximating the state-space to much lower dimensions
The low dimension solution space, called as the Proper Orthogonal Decomposition (POD) subspace, is generated and a reduced system for the problem at hand is constructed by projecting onto the low dimension solution space; in the online stage, on the other hand, an approximated solution is obtained by integrating the reduced dynamical system
The column Goff means the number of initial global POD basis for each time instants, while the column Loff means the number of initial local basis functions, and the column Ladd means the number of local basis added to solve the residual problem (9)
Summary
Reduced order models (ROM) aims at reducing the computational complexity and, in turn, the simulation time of large-scale dynamical systems by approximating the state-space to much lower dimensions. For nonlinear systems, as in the case of flow in heterogeneous porous media, one cannot perform such projection without having to recompute the operators for every time step In this case, the best choice is to compute the projection matrices offline and perform the projection and integration of the reduced model online. In [11,35], POD-DEIM model reduction was used to solve a two-phase flow problem with gravity These methods can only deal with cases that the variation in inputs is within a certain range, i.e., the offline procedure should be able to anticipate almost all the the dynamics of the online system.
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